Internal problem ID [9804]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-5. Equations containing
combinations of trigonometric functions.
Problem number: 49.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\lambda \sin \left (\lambda x \right ) y^{2}-a \left (\cos ^{n}\left (\lambda x \right )\right ) y+a \left (\cos ^{n -1}\left (\lambda x \right )\right )=0} \end {gather*}
✗ Solution by Maple
dsolve(diff(y(x),x)=lambda*sin(lambda*x)*y(x)^2+a*cos(lambda*x)^n*y(x)-a*cos(lambda*x)^(n-1),y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 91.771 (sec). Leaf size: 467
DSolve[y'[x]==\[Lambda]*Sin[\[Lambda]*x]*y[x]^2+a*Cos[\[Lambda]*x]^n*y[x]-a*Cos[\[Lambda]*x]^(n-1),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\frac {a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}}{(n+1) \lambda }\right ) \tan (\lambda K[1]) \left (-a \csc (\lambda K[1]) \cos ^n(\lambda K[1])+a \csc (\lambda K[1]) y(x) \cos ^{n+1}(\lambda K[1])+\lambda y(x)^2 \cos (\lambda K[1])\right )}{(\cos (\lambda K[1]) y(x)-1)^2}dK[1]+\int _1^{y(x)}\left (\frac {\exp \left (-\frac {a \cos ^{n+1}(x \lambda ) \csc (x \lambda ) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(x \lambda )\right ) \sqrt {\sin ^2(x \lambda )}}{(n+1) \lambda }\right )}{(\cos (x \lambda ) K[2]-1)^2}-\int _1^x\left (\frac {2 \exp \left (-\frac {a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}}{(n+1) \lambda }\right ) \left (-a \csc (\lambda K[1]) \cos ^n(\lambda K[1])+a \csc (\lambda K[1]) K[2] \cos ^{n+1}(\lambda K[1])+\lambda K[2]^2 \cos (\lambda K[1])\right ) \sin (\lambda K[1])}{(\cos (\lambda K[1]) K[2]-1)^3}-\frac {\exp \left (-\frac {a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}}{(n+1) \lambda }\right ) \left (a \csc (\lambda K[1]) \cos ^{n+1}(\lambda K[1])+2 \lambda K[2] \cos (\lambda K[1])\right ) \tan (\lambda K[1])}{(\cos (\lambda K[1]) K[2]-1)^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]